Papers by Jeffrey McGowan
This paper means to correct an error by the authors for the composite q case in the paper "L... more This paper means to correct an error by the authors for the composite q case in the paper "Lens Spaces, Isospectral on Forms but not on Functions", published in LMS J. Comput. Math. 9 (2006), 270-286. All calculations and examples presented in <cit.> for prime q remain valid, and we include detailed calculations below justifying this. Our original mistake was to conclude that Formula (3.11) <cit.> remained true for all q when in fact it is only valid if q is prime. This means formulas (3) and (4) in <cit.> must be reworked to account for complications when q is composite.
We investigate the size of the embedded regular tree rooted at a vertex in a d regular random gra... more We investigate the size of the embedded regular tree rooted at a vertex in a d regular random graph. We show that almost always, the radius of this tree will be 1/2 n, where n is the number of vertices in the graph. And we give an asymptotic estimate for Gauss' Hypergeometric Function.
We consider a family of manifolds with a class of degenerating warped product metrics g_ϵ=ρ(ϵ,t)^... more We consider a family of manifolds with a class of degenerating warped product metrics g_ϵ=ρ(ϵ,t)^2adt^2 +ρ(ϵ,t)^2bds_M^2, with M compact, ρ homogeneous degree one, a < -1 and b > 0. We study the Laplace operator acting on L^2 differential p-forms and give sharp accumulation rates for eigenvalues near the bottom of the essential spectrum of the limit manifold with metric g_0.
Harmonic representatives for cuspidal cohomology classes
Number Theory, Analysis and Geometry, 2011
ABSTRACT We give a construction of harmonic differentials that uniquely represent cohomology clas... more ABSTRACT We give a construction of harmonic differentials that uniquely represent cohomology classes of a non-compact Riemann surface of finite topology. We construct these differentials by cutting off all cusps along horocycles and solving a suitable boundary value problem on the truncated surface. We then pass to the limit as the horocycle in each cusp recedes to infinity.
Geometriae Dedicata, 2002
Let G be a convex co-compact, torsion-free, discrete group of isometries of real hyperbolic space... more Let G be a convex co-compact, torsion-free, discrete group of isometries of real hyperbolic space Hn+1. We compute the asymptotics of the counting function for closed geodesics in homology classes for the quotient manifold X = G\Hn+1, under the assumption that H1(X, Z) is infinite. Our results imply asymptotic equipartition of geodesics in distinct homology classes.
Transactions of the American Mathematical Society, 1995
We consider a sequence ( M n ) n = 1 ∞ ({M_n})_{n = 1}^\infty of compact hyperbolic manifolds con... more We consider a sequence ( M n ) n = 1 ∞ ({M_n})_{n = 1}^\infty of compact hyperbolic manifolds converging to a complete hyperbolic manifold M 0 {M_0} with cusps. The Laplace operator acting on the space of L 2 {L^2} differential forms on M 0 {M_0} has continuous spectrum filling the half-line [ 0 , ∞ ) [0,\infty ) . One expects therefore that the spectra of this operator on M n {M_n} accumulate to produce the continuous spectrum of the limiting manifold. We prove that this is the case and obtain a sharp estimate of the rate of accumulation.
An elementary proof that random Fibonacci sequences grow exponentially
ABSTRACT. We consider random Fibonacci sequences given by xn+1 = ±βxn +xn−1. Viswanath ([4]), fol... more ABSTRACT. We consider random Fibonacci sequences given by xn+1 = ±βxn +xn−1. Viswanath ([4]), following Furstenberg ([2]) showed that when β = 1, limn→ ∞ |xn | 1/n = 1.13..., but his proof involves the use of floating point computer calculations. We give a completely elementary proof that 1.23375 ≥ (E(|xn|)) 1/n ≥ 1.12095 where E(|xn|) is the expected value for the absolute value of the nth term in a random Fibonacci sequence. We compute this expected value using recurrence relations which bound the sum of all possible nth terms for such sequences. In addition, we give upper ands lower bounds for the second moment of the |xn|. Finally, we consider the conjecture of Embree and Trefethen ([1]), derived using computational calculations, that for values of β < 0.702585 such sequences decay. We show that as β decreases, the critical value where growth can change to decay is in fact 1 √. 2 Consider the random Fibonacci sequence generated by the recursive rule xn+1 = ±xn + xn−1. Clearly...
The length of closed geodesics on random Riemann Surfaces
Short geodesics are important in the study of the geometry and the spectra of Riemann surfaces. B... more Short geodesics are important in the study of the geometry and the spectra of Riemann surfaces. Bers ’ theorem gives a global bound on the length of the first 3g−3 geodesics. We use the construction of Brooks and Makover of random Riemann surfaces to investigate the distribution of short (< log(g)) geodesics on a random Riemann surfaces. We calculate the expected value of the shortest geodesic, and show that if one orders prime non-intersecting geodesics by length γ1 ≤ γ2 ≤ · · · ≤ γi,..., then for fixed k, if one allows the genus to go to infinity, the length of γk is independent of the genus. 1
arXiv: Differential Geometry, 2019
This paper means to correct an error by the authors for the composite $q$ case in the paper "... more This paper means to correct an error by the authors for the composite $q$ case in the paper "Lens Spaces, Isospectral on Forms but not on Functions", published in LMS J. Comput. Math.} 9 (2006), 270-286. All calculations and examples presented in \cite{GM} for prime $q$ remain valid, and we include detailed calculations below justifying this. Our original mistake was to conclude that Formula (3.11) \cite[p. 399]{Ikeda} remained true for all $q$ when in fact it is only valid if $q$ is prime. This means formulas (3) and (4) in \cite{GM} must be reworked to account for complications when $q$ is composite.
Bounds on Accumulation Rates of Eigenvalues on Manifolds with Degenerating Metrics
We consider a family of manifolds with a class of degenerating warped product metrics $g_\epsilon... more We consider a family of manifolds with a class of degenerating warped product metrics $g_\epsilon=\rho(\epsilon,t)^{2a}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$, with $M$ compact, $\rho$ homogeneous degree one, $a \le -1$ and $b > 0$. We study the Laplace operator acting on $L^{2}$ differential $p$-forms and give sharp accumulation rates for eigenvalues near the bottom of the essential spectrum of the limit manifold with metric $g_{0}$.
We investigate the size of the embedded regular tree rooted at a vertex in a d regular random gra... more We investigate the size of the embedded regular tree rooted at a vertex in a d regular random graph. We show that almost always, the size of this tree will be 1 logn, where n is the number of vertices in the graph. We use this to give an asymptotic estimate for Gauss’ Hypergeometric Function.
Small eigenvalues of the Hodge Laplacian for three-manifolds with pinched negative curvature
Contemporary Mathematics, 1999
... Basic theory. Texts in Applied Mathemat-ics 23. Springer-Verlag, 1996. [26] W. THURsTON. The ... more ... Basic theory. Texts in Applied Mathemat-ics 23. Springer-Verlag, 1996. [26] W. THURsTON. The geometry and topology of 3-manifolds. Princeton. ... Graduate Texts in Mathematics 94. Springer-Verlag, 1983. RUTH CORNET. TExAs TECH UNIVERsITY. ...
We consider a family of manifolds with a class of degenerating warped product metrics $g_\epsilon... more We consider a family of manifolds with a class of degenerating warped product metrics $g_\epsilon=\rho(\epsilon,t)^{2a}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$, with $M$ compact, $\rho$ homogeneous degree one, $a \le -1$ and $b > 0$. We study the Laplace operator acting on $L^{2}$ differential $p$-forms and give sharp accumulation rates for eigenvalues near the bottom of the essential spectrum of the limit manifold with metric $g_{0}$.
Mathematische Annalen, 1993
LMS Journal of Computation and Mathematics, 2006
We study thep-form spectrum of the Laplace-Beltrami operator acting on lens spaces as considered ... more We study thep-form spectrum of the Laplace-Beltrami operator acting on lens spaces as considered by Ikeda [Geometry of manifolds(Academic Press, Boston, MA, 1989) 383–417]. Ikeda gave examples of such spaces that are non-isometric but isospectral for allp≤p0. In this paper we exhibit examples of such spaces that are not isometric, and are isospectral for various, but not for all. values ofp. In particular, examples are given of non-isometric lens spaces that are isospectral for some values ofpbut not for the casep= 0.
We investigate the size of the embedded regular tree rooted at a vertex in a $d$ regular random g... more 
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Papers by Jeffrey McGowan